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| Preface | |
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| First-Order Differential Equations and Their Applications | |
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| Introduction to Ordinary Differential Equations | |
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| The Definite Integral and the Initial Value Problem | |
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| The Initial Value Problem and the Indefinite Integral | |
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| The Initial Value Problem and the Definite Integral | |
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| Mechanics I: Elementary Motion of a Particle with Gravity Only | |
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| First-Order Separable Differential Equations | |
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| Using Definite Integrals for Separable Differential Equations | |
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| Direction Fields | |
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| Existence and Uniqueness | |
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| Euler's Numerical Method (optional) | |
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| First-Order Linear Differential Equations | |
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| Form of the General Solution | |
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| Solutions of Homogeneous First-Order Linear Differential Equations | |
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| Integrating Factors for First-Order Linear Differential Equations | |
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| Linear First-Order Differential Equations with Constant Coefficients and Constant Input | |
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| Homogeneous Linear Differential Equations with Constant Coefficients | |
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| Constant Coefficient Linear Differential Equations with Constant Input | |
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| Constant Coefficient Differential Equations with Exponential Input | |
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| Constant Coefficient Differential Equations with Discontinuous Input | |
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| Growth and Decay Problems | |
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| A First Model of Population Growth | |
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| Radioactive Decay | |
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| Thermal Cooling | |
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| Mixture Problems | |
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| Mixture Problems with a Fixed Volume | |
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| Mixture Problems with Variable Volumes | |
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| Electronic Circuits | |
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| Mechanics II: Including Air Resistance | |
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| Orthogonal Trajectories (optional) | |
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| Linear Second- and Higher-Order Differential Equations | |
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| General Solution of Second-Order Linear Differential Equations | |
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| Initial Value Problem (for Homogeneous Equations) | |
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| Reduction of Order | |
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| Homogeneous Linear Constant Coefficient Differential Equations (Second Order) | |
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| Homogeneous Linear Constant Coefficient Differential Equations (nth-Order) | |
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| Mechanical Vibrations I: Formulation and Free Response | |
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| Formulation of Equations | |
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| Simple Harmonic Motion (No Damping, [delta] = 0) | |
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| Free Response with Friction ([delta] > 0) | |
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| The Method of Undetermined Coefficients | |
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| Mechanical Vibrations II: Forced Response | |
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| Friction is Absent ([delta] = 0) | |
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| Friction is Present ([delta] > 0) (Damped Forced Oscillations) | |
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| Linear Electric Circuits | |
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| Euler Equation | |
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| Variation of Parameters (Second-Order) | |
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| Variation of Parameters (nth-Order) | |
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| The Laplace Transform | |
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| Definition and Basic Properties | |
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| The Shifting Theorem (Multiplying by an Exponential) | |
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| Derivative Theorem (Multiplying by t) | |
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| Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions) | |
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| Initial Value Problems for Differential Equations | |
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| Discontinuous Forcing Functions | |
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| Solution of Differential Equations | |
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| Periodic Functions | |
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| Integrals and the Convolution Theorem | |
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| Derivation of the Convolution Theorem (optional) | |
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| Impulses and Distributions | |
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| An Introduction to Linear Systems of Differential Equations and Their Phase Plane | |
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| Introduction | |
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| Introduction to Linear Systems of Differential Equations | |
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| Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix | |
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| Solving Linear Systems if the Eigenvalues are Real and Unequal | |
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| Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues | |
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| Special Systems with Complex Eigenvalues (optional) | |
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| General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Roots | |
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| Eigenvalues and Trace and Determinant (optional) | |
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| The Phase Plane for Linear Systems of Differential Equations | |
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| Introduction to the Phase Plane for Linear Systems of Differential Equations | |
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| Phase Plane for Linear Systems of Differential Equations | |
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| Real Eigenvalues | |
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| Complex Eigenvalues | |
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| General Theorems | |
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| Mostly Nonlinear First-Order Differential Equations | |
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| First-Order Differential Equations | |
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| Equilibria and Stability | |
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| Equilibrium | |
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| Stability | |
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| Review of Linearization | |
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| Linear Stability Analysis | |
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| One-Dimensional Phase Lines | |
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| Application to Population Dynamics: The Logistic Equation | |
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| Nonlinear Systems of Differential Equations in the Plane | |
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| Introduction | |
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| Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane | |
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| Linear Stability Analysis and the Phase Plane | |
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| Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines | |
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| Population Models | |
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| Two Competing Species | |
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| Predator-Prey Population Models | |
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| Mechanical Systems | |
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| Nonlinear Pendulum | |
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| Linearized Pendulum | |
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| Conservative Systems and the Energy Integral | |
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| The Phase Plane and the Potential | |
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| Answers to Odd-Numbered Exercises | |
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| Index | |